{
  "id": "1911.10889",
  "version": "v1",
  "published": "2019-11-25T13:07:36.000Z",
  "updated": "2019-11-25T13:07:36.000Z",
  "title": "A strong renewal theorem for relatively stable variables",
  "authors": [
    "Kohei Uchiyama"
  ],
  "comment": "13 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $X$ be a non-negative random variable that is relatively stable, or what amounts to the same, $\\int_0^x P[X>t]dt \\sim \\ell(x)$ for a slowly varying function $\\ell$. We show that if $X$ is not arithmetic, the renewal function $U(x)$ associated with $X$ satisfies $\\lim_{x\\to\\infty}[U(x+h)-U(x)]\\ell(x) = h$ for $h>0$. An obvious analogue also holds for the arithmetic variable. The extension to not necessarily non-negative $X$ is obtained.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-25T13:07:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "strong renewal theorem",
      "relatively stable variables",
      "arithmetic",
      "renewal function",
      "slowly varying function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}